Trailblazing work of Colding–Minicozzi demonstrated that spheres and cylinders are the only linearly stable singularity models for mean curvature flow [1]}. In particular, the remaining singularity models are unstable so should not generically occur (as conjectured by Huisken). In a previous paper [2]}, the authors introduced new methods to the study of generic mean curvature flow, proving that a large class of singularity models (specifically, singularities with tangent flows modeled on multiplicity one compact or asymptotically conical self-shrinkers) can be indeed avoided by a slight perturbation of the initial conditions.
